Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #323197 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2541 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19676 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,7,5,3,4,8)(9,10)(11,17)(12,20)(13,16)(14,18)(15,19)(21,23)(22,24);(1,3,4,7,5,2,6,8)(10,16,19,18)(11,20,21,24)(12,22)(17,23);(1,2,5,3)(4,7,6,8)(9,11,15,21)(10,17,19,23)(12,18,22,16)(13,20,14,24);(1,4,5,6)(2,7,3,8)(9,12,13,17,15,22,14,23)(10,11,18,20,19,21,16,24)Polycyclic presentation:
2^9|a^2=fi,b^2=eg,c^2=hi,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=ghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 230437);